Investigation of dispersion characteristic in MI- and MII-type single mode optical fibers

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Optics Communications 271 (2007) 413–420

Investigation of dispersion characteristic in MI- and MII-typesingle mode optical fibers

Ali Rostami *, Morteza Savadi-Oskouei

Photonics and Nanocrystals Research Laboratory (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran

Received 13 June 2006; received in revised form 28 August 2006; accepted 20 October 2006

Abstract

Dispersion characteristic of MI and MII type single mode optical fibers is analytically investigated. For this purpose modal anal-ysis of these fibers to obtain possible wave vectors for given system parameters are done. Then using numerical evaluation of thepresented analytical relations, chromatic and waveguide dispersions are calculated. The effects of geometrical and optical parametersof the fibers on dispersion characteristics are investigated. In this analysis, we show that with increase of D (optical parameter) forMI structure the slope of dispersion curve is decreased and the case is reversed for MII structure. Also, with rising of Q (geometricparameter) for MI structure the slope of dispersion curve is decreased and the situation is reversed for MII structure. Finally, weshow that with boosting of R2 for MII structure the slope of dispersion is increased. As a final result, our simulations show thatsmall values for optical parameters are better in MII structure for multi-channel optical communications. In MI structure to obtainsmall dispersion slope, Q can be increased that is easy for fabrication in practice. Finally, Q and R2 are suitable parameters for con-trol of dispersion in the proposed structures.� 2006 Elsevier B.V. All rights reserved.

Keywords: Dispersion characteristic control; Special optical fiber; Zero-dispersion wavelength; Chromatic dispersion

1. Introduction

Optical fiber is basic and important physical elementfor optical communications. Intensity loss and pulsebroadening due to dispersion phenomenon are two mainproblems in optical fiber based communication. Theseparameters are two basic phenomenons against high-speed and broadband communications. Today low lossoptical fibers are available in some specified wavelengthssuch as 1.55 lm that are one of the alternatives for tradi-tional optical carrier. But dispersion has critical effect forhigh-speed communications especially for large channelsdense wavelength division multiplexing (DWDM) sys-tems. Dispersion management was developed so far forcompensation of dispersion in optical fibers. Developed

0030-4018/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2006.10.072

* Corresponding author. Tel./fax: +98 411 3393724.E-mail address: [email protected] (A. Rostami).

structures in this domain have some basic problems thatshow inefficiency of the proposed ideas. Recently, high-bandwidth applications are requested in optical system.So, design of optical fibers including high bandwidth aswell as low and uniform dispersion profile is interestingfor researchers and applied engineers. For this purpose,in this paper two interesting structures for single modeoptical fiber including possibility (easy and practical) fordispersion management is proposed. In this directionthere are more published papers, which we try to reviewsome of them in the following.

In [1], authors analyzed the chromatic dispersion andsome other interesting quantities for one kind of single-ring profile fiber. However, in that paper they couldnot calculate waveguide dispersion because of their pre-sented complex eigenvalue equation. For this reasonsome of the interesting quantities such as cutoff charac-teristic were not calculated. Also, the effect of system

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414 A. Rostami, M. Savadi-Oskouei / Optics Communications 271 (2007) 413–420

geometrical and optical parameters did not be includedin the analysis based on the presented method. Sincethese effects are so critical in optical fiber analysis andthen the presented method from this point of view isinefficient.

In [2], theoretical analysis of coaxial optical fibers waspresented and detailed analysis of dispersion and the effectof system parameters on interesting fiber quantities werenot included.

In [3], fiber including small dispersion slope has beendesigned. In this paper, authors designed an optical fiberincluding flat mode field in core for increasing mode fielddiameter concluding to low nonlinear effect. In this paper,results based on the numerical simulations, were presentedespecially there were not any analytical results describingdispersion quantity. Also, the effects of optical and geomet-rical parameters on dispersion behavior have not beenstudied.

In [4], authors presented a theoretical analysis of RI-and RII single mode optical fibers and some simulationsfor dispersion quantity were presented also. In this paper,authors concentrated on effective area of the proposedstructures and only one typical simulation for dispersionof these structures was discussed.

Also, these authors have published two papers [5,6] andthose discussed complete analysis compared [4] for WI andWII structures. But, in these papers complete analyticalformulation for dispersion were not presented. It is possibleto present complete analytical formulation for some inter-esting quantities in optical fibers for the presentedstructures.

In this paper, we consider two interesting structures MIand MII triple clad single mode fibers. Also, complete ana-lytical formulation for dispersion analysis of thesestructures is investigated. Thus, the effect of system geo-metrical and optical parameters on dispersion quantitybased on the developed analytical and compact formula-tion is done. In this analysis, zero-dispersion wavelengthmanaging using effective geometrical and optical parame-ters is illustrated. Our developed method can illustratethe effects of parameters and show the efficient controllingparameters on dispersion characteristic of the proposedstructures. The organization of the paper is as follows.

Fig. 1. The index of refraction profile for the proposed str

In Section 2, mathematical modeling for complete ana-lytical formulation is presented. Total dispersion analysisfor description of the simulated results is discussed in Sec-tion 3. Simulated results are presented in Section 4 and dis-cussed there. Finally, the paper ends with a shortconclusion.

2. Mathematical modeling

In this section, the following special fiber structures(Fig. 1) for investigation of dispersion characteristic areconsidered. The corresponding refractive index distributionfunction is defined as follows:

nðrÞ ¼

n1; 0 < r < a;

n2; a < r < b;

n3; b < r < c;

n4; c < r;

8>>><>>>:

ð2:1Þ

where r is the radius position of optical fiber. For thesestructures the effective index of refraction is given byneff ¼ b

k0, where b is the propagation wave vector of guided

modes and k0 is the wave number in vacuum. Based on theeffective-refractive index the proposed structures (MI andMII) can be divided to three and two regions of operation,respectively, which is defined as follows:

MI I : n4 < neff < n2; II : n2 < neff < n1;

III : n1 < neff < n3 ð2:2ÞMII I : n4 < neff < n1; II : n1 < neff < n3 ð2:3Þ

For the proposed structures the following table showstransverse component of the field distribution in all regionsof the fibers.

According to the traditional LP approximation in waveequation propagating in optical fibers and boundary condi-tions the characteristic equations for the proposed struc-tures in the defined regions of operation can be derivedas follows:

where Jm, Ym, Im and Km are the Bessel and modified Bes-sel functions. Also, the transversal propagation constantsused in Eqs. (2.4), (2.5), and (2.6) and Table 1 are definedas follows:

uctures with defined parameters: (a) MI and (b) MII.

J mðU 1Þ �J mðU 2Þ �Y mðU 2Þ 0 0 0

0 J mðU 2Þ Y mðU 2Þ �J mðU 3Þ �Y mðU 3Þ 0

0 0 0 J mðU 3Þ Y mðU 3Þ �KmðW 4ÞU 1J 0mðU 1Þ �U 2J 0mðU 2Þ �U 2Y 0mðU 2Þ 0 0 0

0 U 2J 0mðU 2Þ U 2Y 0mðU 2Þ �U 3J 0mðU 3Þ �U 3Y 0mðU 3Þ 0

0 0 0 U 3J 0mðU 3Þ U 3Y 0mðU 3Þ �W 4K 0mðW 4Þ

��

��¼ 0; ðn4 < ne < n2;MIÞ; ð2:4Þ

J mðU 1Þ �ImðW 2Þ �KmðW 2Þ 0 0 0

0 ImðW 2Þ KmðW 2Þ �J mðU 3Þ �Y mðU 3Þ 0

0 0 0 J mðU 3Þ Y mðU 3Þ �KmðW 4ÞU 1J 0mðU 1Þ �W 2I 0mðW 2Þ �W 2K 0mðW 2Þ 0 0 0

0 W 2I 0mðW 2Þ W 2K 0mðW 2Þ �U 3J 0mðU 3Þ �U 3Y 0mðU 3Þ 0


��

��¼ 0; ðn2 < ne < n1;MIÞ;

ðn4 < ne < n1;MIIÞ; ð2:5ÞImðW 1Þ �ImðW 2Þ �KmðW 2Þ 0 0 0

0 ImðW 2Þ KmðW 2Þ �J mðU 3Þ �Y mðU 3Þ 0

0 0 0 J mðU 3Þ Y mðU 3Þ �KmðW 4ÞW 1I 0m W 1ð Þ �W 2I 0m W 2ð Þ �W 2K 0m W 2ð Þ 0 0 0

0 W 2I 0mðW 2Þ W 2K 0mðW 2Þ �U 3J 0mðU 3Þ �U 3Y 0mðU 3Þ 0


��

��¼ 0; ðn1 < ne < n3;MI;MIIÞ;

ð2:6Þ

A. Rostami, M. Savadi-Oskouei / Optics Communications 271 (2007) 413–420 415

U 1 ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0n21 � b2

q; W 1 ¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � k2

0n21

q; ð2:7Þ

U 2 ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0n22 � b2

q; W 2 ¼ a


0n22

q; ð2:8Þ

U 3 ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0n23 � b2

q; W 4 ¼ c


0n24

q; ð2:9Þ

U 3 ¼1

PU 3; W 2 ¼

PQ

W 2; U 2 ¼PQ

U 2; ð2:10Þ

Table 1Transversal field distribution for different regions and effective-refractive inde

Region (from core to cladding) n4 < neff < n2 (MI) n2 <

IAJ m

U 1ra

� �AJ

IIBJ m

U 2ra

� �þ CY m

U 2ra

� �BI

IIIDJ m

U 3rb

� �þ EY m

U 3rb

� �DJ

IVFKm

W 4rc

� �FK

where P and Q are the geometrical parameters defined asfollows:

P ¼ bc; Q ¼ a

c: ð2:11Þ

Also, the optical parameters are defined as

R1 ¼n3 � n1

n3 � n2

; R2 ¼n2 � n4

n3 � n2

: ð2:12Þ

x ranges

neff < n1 (MI) n4 < neff < n1 (MII) n1 < neff < n3 (MI, MII)

m

U 1ra

� �AIm

W 1ra

� �

mW 2r

a

� �þ CKm

W 2ra

� �BIm

W 2ra

� �þ CKm

W 2ra

� �

m

U 3rb

� �þ EY m

U 3rb

� �DJ m

U 3rb

� �þ EY m

U 3rb

� �

m

W 4rc

� �FKm

W 4rc

� �

1 e 3


It should be pointed out that the positive and negativesigns of R2 are corresponds to MI and MII types, respec-tively. For evaluating of the index of refraction differencebetween core and cladding the following definition is done

D ¼ n23 � n2

4

2n24

� n3 � n4

n4

: ð2:13Þ

Also, the normalized frequency and propagation parame-ters are defined as follows:

V ¼ k0bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

3 � n24

q; ð2:14Þ

B ¼ ðb=k0Þ2 � n24

n23 � n2

4

¼ 1� U 3

V

� �2

: ð2:15Þ

Here, based on the formulation presented in Eqs. (2.4),(2.5) and (2.6) and numerical methods guiding modesand propagation wave vectors can be calculated. Also,using basic relations in waveguide and material dispersionsthe total dispersion parameter for the introduced optical fi-bers is calculated in the following section.

3. Dispersion analysis for the proposed structures

In this section, dispersion analysis based on derived rela-tions in previous section including waveguide and materialdispersions (single mode fibers) is done. The total disper-sion relation for the introduced single mode fibers is asfollows:

D ¼ � kc

d2n4

dk21þ D

dðVBÞdV

� �� N 4

cDk

Vd2ðVBÞ

dV 2; ð3:1Þ

where N 4 ¼ n4 � k dn4

dk is the group index of the outer clad-ding layer. The Sellmeier formula [7] can be used for calcu-

lation of material dispersion dn4

dk and d2n4

dk2

� . Also, for

evaluation of Eq. (3.1), dðVBÞdV and d2ðVBÞ

dV 2 should be calculated.

For this purpose Eq. (2.15) is used and Eqs. (3.2) and (3.3)are concluded

dðVBÞdV

¼ 1þ U 3

V

� �2

1� 2VU 3

� �dU 3

dV

� �; ð3:2Þ

Vd2ðVBÞ

dV 2¼ �2

dU 3

dV� U 3

V

� �2

� 2U 3

d2U 3

dV 2: ð3:3Þ

For complete evaluation of total dispersion, calculation ofEq. (3.2) and Eq. (3.3) are necessary. For this task, calcu-lation of U3 and it derivatives are critical in this step. Forcalculation of U3 the propagation wave vectors should becalculated, which had done using the determinants, whichwas obtained based on application of boundary conditionsto the Maxwell equation (Eqs. (2.4), (2.5) and (2.6)). Forinvestigation of the first derivative of U3, first derivativeof the mentioned above determinants must be done. Inthe analytically calculated derivative of determinants thereare more parameters dependent to the propagation wave

vector. In this step, all transversal propagation constantscan be expressed in terms of U3 using Eqs. (2.7), (2.8)and (2.9). In the following, the proposed method is per-formed in detail. For the following region of operationEq. (2.4) is eigenvalue equation and considered:

ðn4 < ne < n2;MIÞ: ð3:4Þ

So, for calculation of first derivative of U3, one shouldapply derivative operator on both side of Eq. (2.4). Byapplying derivative operator on this determinant,

dU1;2;3

dVand dW 4

dV are appeared in this relation. These terms can beexpressed in terms of dU3

dV and U3 using Eqs. (2.7)–(2.9) as

dU 1

dV¼ Q

P

� �2 1

U 1

U 3

dU 3

dV� q1V

� �; ð3:5Þ

dU 2

dV¼ Q

P

� �2 1

U 2

U 3

dU 3

dV� q2V

� �; ð3:6Þ

dW 4

dV¼ 1

P

� �2 1

W 4

�U 3

dU 3

dVþ V

� �: ð3:7Þ

Now, using these relations and first derivative of determi-nant appeared in Eq. (2.4), the final relation for dU3

dV canbe calculated as follows:

dU 3

dV¼

oFoU1

QP

�2q1

VU1þ oF

oU2

QP

�2q2

VU2

� � oF

oW 4

1P

�2 VW 4

� oFoU1

QP

�2 U3

U1þ oF

oU2

QP

�2 U3

U2

� þ oF

oU3� oF

oW 4

1P

�2 U3

W 4

� ;ð3:8Þ

where q1 ¼n2

3�n2

1

n23�n2

4

ffi n3�n1

n3�n4, q2 ¼

n23�n2

2

n23�n2

4

ffi n3�n2

n3�n4and F is determi-

nant in Eq. (2.4). For the following second region of oper-ation Eq. (2.5) is eigenvalue equation:

ðn2 < ne < n1;MIÞ;ðn4 < ne < n1;MIIÞ:

ð3:9Þ

So, for calculation of dU3

dV , one should apply derivative oper-ator on both side of Eq. (2.5). By applying derivative oper-

ator on this determinant,dU1;3

dV anddW 2;4

dV are appeared in thisrelation. So, these terms can be expressed in terms of dU3

dVand U3 as follows and appeared in Eqs. (3.5)–(3.7):

dW 2

dV¼ Q

P

� �21

W 2

�U 3

dU 3

dVþ q2V

� �; ð3:10Þ

Now, using these relations and first derivative of determi-nant appeared in Eq. (2.5), the final relation for first deriv-ative of U3 can be calculated as follows:

dU 3

dV¼

oFoU1

QP

�2q1

VU1� oF

oW 2

QP

�2q2

VW 2

� � oF

oW 4

1P

�2 VW 4

� oFoU1

QP

�2 U3

U1� oF

oW 2

QP

�2 U3

W 2

� þ oF

oU3� oF

oW 4

1P

�2 U3

W 4

� ;ð3:11Þ

where F is determinant appeared in Eq. (2.5). Finally, forthe following third region of operation Eq. (2.6) is eigen-value equation:

ðn < n < n ;MI;MIIÞ: ð3:12Þ


So, for calculation of dU3

dV , one should apply derivative oper-ator on both side of Eq. (2.5). By applying derivative oper-

ator on this determinant, dU3

dV anddW 1;2;4

dV are appeared in this

relation. So, these terms can be expressed in terms of dU3

dV

and U3 based on basic relations proposed in Eqs. (2.5)–(2.7), (3.7), (3.10) and as follows:

dW 1

dV¼ Q

P

� �21

W 1

�U 3d3

dVþ q1V

� �: ð3:13Þ

Now, using these relations and first derivative of determi-nant appeared in Eq. (2.6), the final relation for dU3

dV canbe calculated as follows:

dU 3

dV¼

oFoW 1

QP

�2q1

VW 1þ oF

oW 2

QP

�2q2

VW 2

� þ oF

oW 4

1P

�2 VW 4

� oF

oW 1

QP

�2 U3

W 1þ oF

oW 2

QP

�2 U3

W 2

� � oF

oU3þ oF

oW 4

1P

�2 U3

W 4

� ;ð3:14Þ

where F is determinant appeared in Eq. (2.6). For complet-ing the calculation of total dispersion, second derivative ofU3 should be calculated. For this purpose, second deriva-tive of the mentioned above determinants are necessary.For do that, we treat such as first derivative case. Three re-gion of operation defined in Section 2 are considered.

For the following region of operation Eq. (2.4) isconsidered:

ðn4 < ne < n2;MIÞ: ð3:15ÞSo, for calculation of second derivative of U3, one shouldapply derivative operator on both side of Eq. (2.4) twotimes. By applying derivative operator on this determinant,d2U1;2;3

dV 2 and d2W 4

dV 2 are appeared in this relation. So, these terms

can be expressed in terms of d2U3

dV 2 ;dU3

dV and U3 as follows:

d2U 1

dV 2¼ � 1

U 1

dU 1

dV

� �2

þ QP

� �21

U 1

� dU 3

dV

� �2

þ U 3

d2U 3

dV 2� q1

" #; ð3:16Þ

d2U 2

dV 2¼ � 1

U 2

dU 2

dV

� �2

þ QP

� �21

U 2

� dU 3

dV

� �2

þ U 3

d2U 3

dV 2� q2

" #; ð3:17Þ

d2W 4

dV 2¼ � 1

W 4

dW 4

dV

� �2

þ 1

P

� �21

W 4

� � dU 3

dV

� �2

� U 3d2U 3

dV 2þ 1

" #: ð3:18Þ

Now, using these relations and second derivative of deter-minant appeared in Eq. (2.4), the final relation for secondderivative of U3 can be calculated as follows:

d2U 3

dV 2¼ � X 1 þ X 2 þ X 3 þ X 4

oFoU1

QP

�2 U3

U1þ oF

oU2

QP

�2 U3

U2

� þ oF

oU3� oF

oW 4

1P

�2 U3

W 4

� ;ð3:19Þ

where

X 1 ¼o2F

oU 21

dU 1

dV

� �2

þ o2F

oU 22

dU 2

dV

� �2

þ o2F

oU 23

dU 3

dV

� �2

þ o2F

oW 24

dW 4

dV

� �2

þ 2o2F

oU 2oU 1

dU 1

dV

� ��dU 2

dV

� �

þ o2FoU 3oU 1

dU 1

dV

� �dU 3

dV

� �þ o2F

oW 4oU 1

dU 1

dV

� �dW 4

dV

� �

þ o2F

oU 2oU 3

dU 3

dV

� �dU 2

dV

� �þ o

2FoW 4oU 2

dW 4

dV

� �dU 2

dV

� �

þ o2FoW 4oU 3

dW 4

dV

� �dU 3

dV

� ��; ð3:20Þ

X 2 ¼oFoU 1

� 1

U 1

dU 1

dV

� �2

þ QP

� �21

U 1

� �(

� dU 3

dV

� �2

� q1

" #); ð3:21Þ

X 3 ¼oFoU 2

� 1

U 2

dU 2

dV

� �2

þ QP

� �21

U 2

� �(

� dU 3

dV

� �2

� q2

" #); ð3:22Þ

X 4 ¼oFoW 4

� 1

W 4

dW 4

dV

� �2

þ 1

P

� �21

W 4

� �(

� � dU 3

dV

� �2

þ 1

" #)ð3:23Þ

and F is determinant appeared in Eq. (2.4). For the follow-ing region of operation Eq. (2.5) is eigenvalue equation:

ðn2 < ne < n1;MIÞ;ðn4 < ne < n1;MIIÞ:

ð3:24Þ

So, for calculation of second derivative of U3, one shouldapply derivative operator on both side of Eq. (2.5) twotimes. By applying derivative operator on this determinant,d2U1;3

dV 2 andd2W 2;4

dV 2 are appeared in this relation. So, these terms

can be expressed in terms of d2U3

dV 2 ;dU3

dV and U3 as follows:

d2W 2

dV 2¼ � 1

W 2

dW 2

dV

� �2

þ QP

� �2

� 1

W 2

� dU 3

dV

� �2

� U 3

d2U 3

dV 2þ q2

" #: ð3:25Þ


d2U 3

dV 2¼ � X 1 þ X 2 þ X 3 þ X 4

oFoU1

QP

�2 U3

U1� oF

oW 2

QP

�2 U3

W 2

� þ oF

oU3� oF

oW 4

1P

�2 U3

W 4

� ;ð3:26Þ

where


X 1 ¼o2F

oU 21

dU 1

dV

� �2

þ o2F

oW 22

dW 2

dV

� �2

þ o2F

oU 23

dU 3

dV

� �2

þ o2F

oW 24

dW 4

dV

� �2

þ 2o2F

oW 2oU 1

dU 1

dV

� �dW 2

dV

� ��

þ o2FoU 3oU 1

dU 1

dV

� �dU 3

dV

� �þ o2F

oW 4oU 1

dU 1

dV

� �dW 4

dV

� �

þ o2F

oW 2oU 3

dU 3

dV

� �dW 2

dV

� �þ o

2FoW 4oW 2

dW 4

dV

� �dW 2

dV

� �

þ o2FoW 4oU 3

dW 4

dV

� �dU 3

dV

� ��; ð3:27Þ

X 2 ¼oFoU 1

� 1

U 1

dU 1

dV

� �2

þ QP

� �21

U 1

� �(

� dU 3

dV

� �2

� q1

" #); ð3:28Þ

X 3 ¼oFoW 2

� 1

W 2

dW 2

dV

� �2

þ QP

� �21

W 2

� �(

� � dU 3

dV

� �2

þ q2

" #); ð3:29Þ

X 4 ¼oFoW 4

� 1

W 4

dW 4

dV

� �2

þ 1

P

� �21

W 4

� �(

� � dU 3

dV

� �2

þ 1

" #); ð3:30Þ

and F is determinant appeared in Eq. (2.5). For the follow-ing region of operation Eq. (2.6) is eigenvalue equation:

ðn1 < ne < n3;MI;MIIÞ: ð3:31Þ

So, for calculation of second derivative of U3, one shouldapply derivative operator on both side of Eq. (2.6) twotimes. By applying derivative operator on this determi-

nant, d2U3

dV 2 andd2W 1;2;4

dV 2 are appeared in this relation. So,

these terms can be expressed in terms of d2U3

dV 2 ;dU3

dV andU3 as follows:

d2W 1

dV 2¼ � 1

W 1

dW 1

dV

� �2

þ QP

� �2

� 1

W 1

� dU 3

dV

� �2

� U 3

d2U 3

dV 2þ q1

" #: ð3:32Þ


d2U 3

dV 2¼ � X 1 þ X 2 þ X 3 þ X 4

� oFoW 1

QP

�2 U3

W 1� oF

oW 2

QP

�2 U3

W 2

� þ oF

oU3� oF

oW 4

1P

�2 U3

W 4

� ;ð3:33Þ

where

X 1 ¼o2F

oW 21

dW 1

dV

� �2

þ o2F

oW 22

dW 2

dV

� �2

þ o2F

oU 23

dU 3

dV

� �2

þ o2F

oW 24

dW 4

dV

� �2

þ 2o2F

oW 2oW 1

dW 1

dV

� �dW 2

dV

� ��

þ o2FoU 3oW 1

dW 1

dV

� �dU 3

dV

� �þ o2F

oW 4oW 1

dW 1

dV

� �dW 4

dV

� �

þ o2FoW 2oU 3

dU 3

dV

� �dW 2

dV

� �

þ o2FoW 4oW 2

dW 4

dV

� �dW 2

dV

� �þ o2F

oW 4oU 3

dW 4

dV

� �dU 3

dV

� ��;

ð3:34Þ

X 2 ¼oFoW 1

� 1

W 1

dW 1

dV

� �2

þ QP

� �21

W 1

� �(

� � dU 3

dV

� �2

þ q1

" #); ð3:35Þ

X 3 ¼oFoW 2

� 1

W 2

dW 2

dV

� �2

þ QP

� �21

W 2

� �(

� � dU 3

dV

� �2

þ q2

" #); ð3:36Þ

X 4 ¼oFoW 4

� 1

W 4

dW 4

dV

� �2

þ 1

P

� �21

W 4

� �(

� � dU 3

dV

� �2

þ 1

" #); ð3:37Þ

and F is determinant appeared in Eq. (2.6). Now, based onanalytically derived relations for first and second deriva-tives of U3, calculation of Eq. (3.1) can be done and totaldispersion is extracted. For realization of the presented the-ory, we calculated the presented determinants appeared inEqs. (2.4), (2.5), (2.6) and the propagation vector is ex-tracted. Also, based on the extracted wave vector and thepresented analytical relations total dispersion can be ob-tained and our simulated results will be appeared in the fol-lowing section.

4. Simulation results and discussion

In this section, based on the presented analytical rela-tions the effect of geometrical and optical parameters ondispersion quantity is investigated. In this simulation thepositive and negative signs of optical parameter R2 showMI and MII structures, respectively. In all of the presentedsimulations the wavelength and dispersion are expressed interms of lm and s/m2, respectively. The effect of opticalparameter (D) on dispersion behavior for MII structure isdemonstrated in Fig. 2.

It is shown that with increase of D, the zero-dispersionwavelength is increased. Also, with increase of index ofrefraction difference the slope of dispersion curve versus

-3

-2

-1

0

1

2

3

4x 10

-5

Dis

per

sio

n

Q=0.1Q=0.2Q=0.3

R1=0.5, R2=-0.2, Δ=7e-3P=0.8, a=2.5um,

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-3

-2

-1

0

1

2

3x 10

-5

wavelength

Dis

per

sio

n

Δ= 3e-3

Δ= 5e-3

Δ=7e-3

Δ=1e-2

R1=0.5,R2=-0.3,P=0.8,Q=0.2,a=2.5um

Fig. 2. Dispersion vs. wavelength for MII structure with D as parameter.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-3

-2

-1

0

1

2

3

4x 10

-5

wavelength

Dis

per

sio

n

R2=-0.4R2=-0.2R2=0.3

R1=0.5, Δ=5e-3,Q=0.3,P=0.8,a=2.5um

Fig. 4. Dispersion vs. wavelength for both structures with R2 asparameter.


wavelength is decreased. Therefore, increase of D in largewavelengths has small effect on zero-dispersion wavelengthshifting, but for small wavelength the effect is critical. Theeffect of optical parameter (D) on dispersion characteristicfor MI structure is investigated in Fig. 3. It is shown thatwith increase of D, the zero-dispersion wavelength isdecreased. Also, with increase of index of refraction differ-ence the slope of dispersion curve versus wavelength isdecreased. In this case, the effect of the increase of D on dis-persion curve for small wavelength is critical compared tolarge wavelengths.

Effect of another optical parameter (R2) on dispersioncharacteristic is illustrated in Fig. 4. It is shown that withdecrease of R2 the zero-dispersion wavelength is increasedand shifted to higher wavelengths. Also, the slope of dis-persion curve is decreased with decrease of this parameter.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9wavelength

Fig. 5. Dispersion vs. wavelength for MII structure with Q as parameter.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-3

-2

-1

0

1

2

3

4x 10

-5

wavelength

Dis

per

sio

n

Δ= 3e-3

Δ= 5e-3

Δ= 7e-3

R1=0.5,R2=0.3,P=0.8,Q=0.3, a=2.5um

Fig. 3. Dispersion vs. wavelength for MI structure with D as parameter.

The effect of geometrical parameter (Q) on dispersionbehavior of MII structure is investigated in Fig. 5. Simu-lated result (R2 = �0.2) shows that with decrease of Q

the zero-dispersion wavelength is decreased. Also, it isshown that for small Q decreasing in zero-dispersion wave-length is critical and considerable. As a result, with risingof Q the slope of dispersion is increased.

The effect of geometrical parameter (Q) on dispersionbehavior of MI structure is investigated in Fig. 6. Simula-tion shows that with decrease of Q the zero-dispersionwavelength is increased. Also, it is shown that for smallQ value, decrease in zero-dispersion wavelength is criticaland considerable.

In this section, effects of optical and geometrical param-eters on dispersion characteristics of the proposed struc-tures have been considered and shown that MI and MII

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-3

-2

-1

0

1

2

3

4x 10

-5

wavelength

Dis

per

sio

n

Q=0.3

Q=0.4

Q=0.5

R1=0.5, R2=0.3, P=0.8a=2.5um, Δ=7e-3

Fig. 6. Dispersion vs. wavelength for MI structure with Q as parameter.


structures can be used as optical fibers including controlla-ble dispersion characteristics from zero-dispersion wave-length and slope of dispersion point of views.

5. Conclusion

In this paper, dispersion analysis for the proposedspecial structures for optical fibers has been done. In thisanalysis, we have presented analytical relations for incor-

porating dispersion behavior of the proposed structures.We have shown that, based on developed analytical rela-tions, zero-dispersion wavelength can be controlled by geo-metrical and optical parameters. The effect of systemparameters on dispersion characteristic was investigated.Based on the proposed relations, optimum structures canbe developed for optimum dispersion treatment. Our simu-lations have shown that for MI structure with increase ofthe geometrical parameter Q, the slope of dispersion curveis decreased, so this parameter is important for managingof dispersion characteristic. On the other hand for broadband communication (small slope dispersion) this structurewill be suitable alternative. For MII structure, we haveshown that with decrease of optical parameter R2 the slopeof dispersion curve is decreased. So, for broad band appli-cations based on this structure small R2 is better, that iseasily realizable in practice.

References

[1] H.T. Hattori, A. Safaei-Jazi, Applied Optics 37 (1998) 3190.[2] F.D. Nunes, C.A. de Souza Melo, Applied Optics 35 (1996)

388.[3] R.K. Varshney, A.K. Ghatak, I.C. Goyal, S.C. Antony, Optical Fiber

Technology 9 (2003) 189.[4] X. Tian, X. Zhang, Optics Communications 230 (2004) 105.[5] X. Zhang, X. Tian, Optics & Laser Technology 35 (2003) 237.[6] X. Zhang, X. Wang, Optics & Laser Technology 37 (2005) 167.[7] A. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cambridge

University Press, 2002.

Documents

Investigation of dispersion characteristic in MI- and MII-type single mode optical fibers